The stationary point for $ y=(2+x)e^{-x} $ is:

Question

BlackTom AI · Accepted Answer

Restating the problem: determine the stationary point(s) of the function y = (2 + x) e^{-x}.

First, I compute the derivative using the product rule on (2 + x) and e^{-x}:
- The derivative of (2 + x) is 1.
- The derivative of e^{-x} is -e^{-x}.
Thus, y' = 1 * e^{-x} + (2 + x) * (-e^{-x}) = e^{-x} - (2 + x) e^{-x}.

Factor out e^{-x} (which is never zero for any real x): y' = e^{-x} [1 - (2 + x)] = e^{-x} (-1 - x).

To find stationary points, set y' = 0. Since e^{-x} ≠ 0 for all x, we require -1 - x = 0, which yields x = -1.

Next, compute the corresponding y-coordinate by substituting x = -1 into the original function:
y = (2 + (-1)) e^{-(-1)} = 1 * e^{1} = e.

Therefore, the calculations indicate that the coordinate where the slope is zero is at x = -1, with the corresponding y-value equal to e. The result is the ordered pair consisting of these computed values.

If you compare this with any options given, you would look for the point where the x-coordinate is -1 and the y-coordinate is e. Note that the derivative analysis hinges on the fact that e^{-x} never vanishes, which is why the zero of y' comes solely from -1 - x = 0.

In this solution, the key steps were: differentiate via product rule, factor out the nonzero exponential term, set the remaining linear factor to zero, and substitute back to find the y-coordinate.

MUF0091 Mathematics Unit 1 - Semester 1, 2025 12.7 Study Area 3 revision quiz

The stationary point for \( y=(2+x)e^{-x} \) is:

View Explanation

Log in for full answers

Similar Questions

If the function [math: f(x)=x3+4x2+4x] f(x)=x^3+4x^2+4x has a stationary point at (-2,0), the other stationary point is:

If \( f(x)=x^3+ax^2+bx \) has a stationary point at (-2,0), then the values of \( a \) and \( b \) respectively are:

Given that for a function: and stationary points at and On paper, determine the nature of all stationary points of the function, showing all working. AND In this quiz, select "I have done this question on paper" (Answer in quiz: 0 mark) (Written answer: 2 marks)

Given that a function with a single stationary point has:

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler